AP Calculus BC Series and Sequences Practice Problems (2025 Guide)

Step-by-Step Practice Problem 1: Convergence of a Series

Problem: Determine whether the series

∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2}

converges or diverges.

Solution:

1. Recognize this as a p-series with p = 2.
2. Rule: A p-series converges if p > 1.
3. Since p = 2 > 1, the series converges.

Final Answer: The series converges.

Step-by-Step Practice Problem 2: The Ratio Test

Problem: Does the series

∑n=1∞3nn!\sum_{n=1}^{\infty} \frac{3^n}{n!}

converge or diverge?

Solution:

1. Apply the Ratio Test: L=lim⁡n→∞∣an+1an∣L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|
2. Compute: an+1an=3n+1/(n+1)!3n/n!=3n+1.\frac{a_{n+1}}{a_n} = \frac{3^{n+1}/(n+1)!}{3^n/n!} = \frac{3}{n+1}.
3. Take the limit: L=lim⁡n→∞3n+1=0.L = \lim_{n \to \infty} \frac{3}{n+1} = 0.
4. Since L < 1, the series converges absolutely.

Final Answer: The series converges by the Ratio Test.

Step-by-Step Practice Problem 3: Alternating Series

Problem: Does the series

∑n=1∞(−1)nn\sum_{n=1}^{\infty} \frac{(-1)^n}{n}

converge?

Solution:

1. This is the Alternating Harmonic Series.
2. Apply the Alternating Series Test:
   • Terms decrease in absolute value: 1/n decreases.
   • Limit of terms = 0.
3. Both conditions satisfied → series converges.

Final Answer: The series converges conditionally.

Step-by-Step Practice Problem 4: Interval of Convergence

Problem: Find the interval of convergence for the power series

∑n=0∞(x−2)nn⋅3n\sum_{n=0}^{\infty} \frac{(x-2)^n}{n \cdot 3^n}

Solution:

1. Apply the Ratio Test: L=lim⁡n→∞∣(x−2)n+1/((n+1)3n+1)(x−2)n/(n3n)∣=lim⁡n→∞∣x−2∣3⋅nn+1.L = \lim_{n \to \infty} \left| \frac{(x-2)^{n+1}/((n+1)3^{n+1})}{(x-2)^n/(n3^n)} \right| = \lim_{n \to \infty} \frac{|x-2|}{3} \cdot \frac{n}{n+1}.
2. Simplify: L=∣x−2∣3.L = \frac{|x-2|}{3}.
3. Convergence when L < 1 → |x – 2| < 3.
4. Interval: (–1, 5). Must test endpoints:
   • At x = –1: series = ∑ (–3)^n/(n3^n) = ∑ (–1)^n/n → converges (alternating harmonic).
   • At x = 5: series = ∑ 3^n/(n3^n) = ∑ 1/n → diverges (harmonic series).

Final Answer: Interval of convergence = [–1, 5).

Step-by-Step Practice Problem 5: Maclaurin Series Expansion

Problem: Write the Maclaurin series for sin(x) up to the x⁵ term.

Solution:

1. Formula: sin(x) = x – x³/3! + x⁵/5! – …
2. Up to x⁵: sin(x) ≈ x – x³/6 + x⁵/120.

Final Answer:

sin⁡(x)≈x−x36+x5120\sin(x) \approx x - \frac{x^3}{6} + \frac{x^5}{120}

Common Mistakes in Sequences and Series

• Forgetting that nth-term test only proves divergence, not convergence.
• Mixing up absolute vs. conditional convergence.
• Forgetting to check endpoints in interval of convergence problems.
• Misusing Taylor expansions (not aligning with factorials correctly).
• Leaving justifications incomplete on FRQs (always cite the correct convergence test).

Pro Tips for Scoring on Sequences and Series

• Always write the name of the test you’re applying (Ratio Test, Root Test, etc.). The AP readers want explicit justification.
• When working with Taylor/Maclaurin series, memorize expansions for:
  • sin(x), cos(x), e^x, ln(1+x), 1/(1–x).
• Clearly separate steps when testing endpoints.
• Practice writing full sentences: “By the Ratio Test, since L < 1, the series converges absolutely.”
• Use practice to spot which convergence test applies to each series type.

Additional Practice Problems

1. Determine if the series ∑ (2^n / n³) converges or diverges.
2. Find the radius and interval of convergence for ∑ (x^n / n²).
3. Write the first four nonzero terms of the Maclaurin series for e^(–x²).
4. Does ∑ (–1)^n (n / (n²+1)) converge absolutely, converge conditionally, or diverge?
5. Evaluate ∑ (1 / 3^n) from n=1 to infinity.

(Step-by-step solutions are available in RevisionDojo’s AP Calculus BC series library.)

Frequently Asked Questions

1. What types of series and sequences appear on the AP Calculus BC exam?
You’ll see convergence tests, power series, and Taylor/Maclaurin expansions.

2. Do I need to memorize series for the AP exam?
Yes — especially the Maclaurin series for e^x, sin(x), cos(x), ln(1+x), and 1/(1–x).

3. What’s the most important convergence test?
The Ratio Test is the most commonly applied, especially for factorials and exponentials.

4. How can I avoid mistakes with intervals of convergence?
Always test endpoints separately and justify using the correct convergence test.

5. Where can I find AP-style practice series problems with solutions?
Check RevisionDojo’s AP Calculus BC resources for worked examples and exam-style problems.

Conclusion: Mastering Series and Sequences for AP Success

Series and sequences can feel overwhelming, but they follow clear patterns once you practice the main tests. By mastering p-series, ratio tests, alternating series, and Taylor expansions, you’ll be well-prepared for both multiple-choice and FRQs on the AP Calculus BC exam.

The best way to build confidence is through consistent practice. Work through a wide range of convergence problems, explain your reasoning clearly, and memorize the core Maclaurin series expansions. For structured study plans and expert walkthroughs, visit RevisionDojo and take your AP Calculus prep to the next level.